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Mathematics LibreTexts

1.9: Pigeonhole Principle

  • Page ID
    17370
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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    INVESTIGATE!!

    1. Suppose there are n people at a party, with n at least 2. Show that there are two people that have the same number of friends.
    2. Suppose 5 points are selected from inside a \(1\times 1\) square. Prove that two of the points must be within \(\frac{1}{\sqrt{2}}\) of each other.

    Pigeonhole principle

    • Simple version: If n+1 pigeons are placed in pigeonholes, then at least one pigeonhole contains two or more pigeons.
    • General version: If n or more pigeons are placed in k pigeonholes, then at least one pigeonhole contains \(\lceil\frac{n}{k}\rceil\) or more pigeons.

    As a consequence, we can say that there must be two students at Notre Dame whose phone numbers end with the same four digits. Similarly, there must be two non-bald people in Chicago with the exact same number of hairs on their heads.

    Example \(\PageIndex{1}\):

    Seven distinct numbers are selected from the set \(\{1,2,3,\ldots,11\}\). Prove that two of these numbers must sum to 12.

    Example \(\PageIndex{2}\):

    Prove that among any 7 integers there must be two whose difference is divisible by 6.